What can I do to get rid of ugly triangles
After looking at your file the problem is very obvious.

You have a situation where your UV are laying on top of one another. For baking purposes you should never have your UVs position over one another. All islands must have their own space.

Imagine the renderer using those UV information you gave. First it renders A_island_uv portion of the mesh then render B_island_uv portion of the mesh over the previously rendered A_island_uv and no wonder it's a mess.

Check out the UVs after I corrected them the strange triangles are now gone.



FURTHER INSTRUCTIONS
All you need to do, for this scenario is to enter edit mode. Select a vertice on the UV island you want to move .. Hit L and that will select the whole island. Hit G to grab and move the island. Scale them up or down if need be and you are done.